23 The Application of Time–Domain DQM …
287
Obviously, there are N η × N ζ × N τ algebraic equations in the Eq. (23.10).
Consequently, Eq. (23.10) can be rewritten in the block matrix form as Eq. (23.12):
[s] · {Ω} = {F}
(23.12)
where [s] is the stiffness matrix, [Ω], that is to say, [Ω]
=
W (η 1 , ζ 1 , τ 1 ), W (η 1 , ζ 1 , τ 2 ), . . . , W (η N η , ζ N ζ , τ N τ −1 ), W (η N η , ζ N ζ , τ N τ )
,
is unknown dynamic displacement vector to be achieved next step, and {F} is the
generalized load vector coming from the right hand terms of Eq. (23.10).
Similarly, replacing these governing equations of Eq. (23.12) with the boundary
conditions Eqs. (23.11a–h) and initial conditions Eqs. (23.11i–j) for the same nodes of
space–time domain, we can get the following matrix Eq. (23.13) of simply supported
plates subjected to a uniform transverse load
¯
S
· {Ω} = { ¯
F}
(23.13)
where
¯
S
and { ¯
F} are the updated stiffness matrix and generalized load vector,
respectively.
The dynamic displacement vector can be gained by solving Eq. (23.13) and the
dimensional transverse displacement field in continuous space–time domain can be
obtained by Eq. (23.14):
w = δ
N η
m=1
N ζ
n=1
N τ
p=1
γ m (x)χ n (y)ψ p (t)W (η m , ζ n , τ p )
(23.14)
23.3 Numerical Results
To demonstrate the accuracy and efficiency of the adopted time–domain DQM,
there are two examples for studying the forced vibration of simply supported plates
subjected to a transverse uniform load. For the first one, we calculate the transverse
deflection versus time response of a simply supported plate with ¯
m = 0.008 kg/cm
3 ,
E = 1.5 × 10
7 N/cm
2 , L = 60 cm, B = 60 cm, δ = 1 cm, μ = 0.333. When it
is subjected to a uniform suddenly applied load q 1 = 0.3 N/cm
2 , both the initial
deflection and velocity are w 0 = v 0 = 0. For the second one, we load the same
simply supported plate in the above example with a uniform cosine load q 2 = 0.3×
cos(200t) N/cm
2 , and do the similar work.
Note that for the forced vibration of simply supported plates subjected to a transverse load, the Adj. R2 (so–called adjusted coefficient of determination) could be
used for measuring the degree of the fitting curves, for examples, the degree of a w
versus t curve (i.e., the estimated biomass) gained by DQM to fit the another (namely
the observed biomass) gained by analytical solution, and described by following
287
Obviously, there are N η × N ζ × N τ algebraic equations in the Eq. (23.10).
Consequently, Eq. (23.10) can be rewritten in the block matrix form as Eq. (23.12):
[s] · {Ω} = {F}
(23.12)
where [s] is the stiffness matrix, [Ω], that is to say, [Ω]
=
W (η 1 , ζ 1 , τ 1 ), W (η 1 , ζ 1 , τ 2 ), . . . , W (η N η , ζ N ζ , τ N τ −1 ), W (η N η , ζ N ζ , τ N τ )
,
is unknown dynamic displacement vector to be achieved next step, and {F} is the
generalized load vector coming from the right hand terms of Eq. (23.10).
Similarly, replacing these governing equations of Eq. (23.12) with the boundary
conditions Eqs. (23.11a–h) and initial conditions Eqs. (23.11i–j) for the same nodes of
space–time domain, we can get the following matrix Eq. (23.13) of simply supported
plates subjected to a uniform transverse load
¯
S
· {Ω} = { ¯
F}
(23.13)
where
¯
S
and { ¯
F} are the updated stiffness matrix and generalized load vector,
respectively.
The dynamic displacement vector can be gained by solving Eq. (23.13) and the
dimensional transverse displacement field in continuous space–time domain can be
obtained by Eq. (23.14):
w = δ
N η
m=1
N ζ
n=1
N τ
p=1
γ m (x)χ n (y)ψ p (t)W (η m , ζ n , τ p )
(23.14)
23.3 Numerical Results
To demonstrate the accuracy and efficiency of the adopted time–domain DQM,
there are two examples for studying the forced vibration of simply supported plates
subjected to a transverse uniform load. For the first one, we calculate the transverse
deflection versus time response of a simply supported plate with ¯
m = 0.008 kg/cm
3 ,
E = 1.5 × 10
7 N/cm
2 , L = 60 cm, B = 60 cm, δ = 1 cm, μ = 0.333. When it
is subjected to a uniform suddenly applied load q 1 = 0.3 N/cm
2 , both the initial
deflection and velocity are w 0 = v 0 = 0. For the second one, we load the same
simply supported plate in the above example with a uniform cosine load q 2 = 0.3×
cos(200t) N/cm
2 , and do the similar work.
Note that for the forced vibration of simply supported plates subjected to a transverse load, the Adj. R2 (so–called adjusted coefficient of determination) could be
used for measuring the degree of the fitting curves, for examples, the degree of a w
versus t curve (i.e., the estimated biomass) gained by DQM to fit the another (namely
the observed biomass) gained by analytical solution, and described by following